Remark An advantage of this approach is that it sidesteps the issue of representing observables as operators and therefore of constructing the Hilbert spaces on which they presumably act. Indeed, in QFT, where the number of field degrees of freedom are (uncountably) infinite, the Stone-von Neumann theorem does not hold (see e.g., [1]}) and thus there are an infinite number of representations of the canonical commutation relations, all of which are unitarily inequivalent. While the issue is less problematic in a flat space-time, where a unique Poincaré invariant vacuum state can be constructed, it presents a significant problem in curved space-time where there is no Poincaré invariance, and consequently, no unique vacuum state, either. In ED, however, a quantum theory is obtained without encountering these issues which are emblematic of an emphasis on Hilbert space methods. Thus, as we will see in the next chapter, the fact that ED proceeds without appealing to the notion of Hilbert spaces is thus a decided advantage for constructing a quantum theory in the context of a curved space-time.

